An empirical mode decomposition (EMD) method is proposed by Huang N. E. et al. to perform non-stationary and non-linear signal decomposition. The algorithm of signal decomposition can decompose a time-related signal into superposition of a limited number of intrinsic mode functions (IMF) and monotonic functions. Nowadays, there have existed a large number of documents verifying the signal analysis ability of one-dimensional empirical mode decomposition. The two-dimensional empirical mode decomposition has also been applied in image processing, such as image texture analysis, image edge detection and a minority of medical image applications.
After the year of 2000, the two-dimensional empirical mode decomposition is applied to picture processing. The two-dimensional empirical mode has only slight difference from the one-dimensional empirical mode in mathematical theory. Basically, the empirical mode decomposition obtains characteristic waveforms by many times of squeezing with envelope lines of the maximum and minimum values of signals. When applied in the two-dimensional empirical mode, a more complicated envelope surface is used instead of the use of simple envelope lines.
The envelope surface used in the two-dimensional empirical mode is formed by image grids, or by auxiliary of the optimal numerical interpolation. However, the present academic-applied two-dimensional empirical mode decomposition has the following three essential problems to be solved.
First, the maximum and minimum values of a two-dimensional image are difficult to define (such as a saddle-shape problem). Second, the present two-dimensional empirical mode based on continuous data would probably lead to inaccuracy due to the discrete and discontinuous nature of images. Third, the concept of an envelope surface cannot be extended to three or more than three dimensional empirical mode.
Regarding the definition of extrema, the extrema of a signal includes maxima and minima. In addition to the conventional definition using signal intensity, the signal curvature is used to define extrema of the signal. However, in the prior-art technology, no discussion is provided about the definition of signal extrema in the case that the signal has the dimension more than one. Further, the signal has usually two, or three, or even four dimensions in the practical applications, such as ultrasonic images, CT images and 4D-ultrasonic images.
The empirical mode decomposition is implemented by a number of iterations in a sifting procedure. An important process in the sifting procedure is the construction of envelope functions. When the data to be analyzed is one-dimensional, the well-known methods of envelope-function construction include a cubic spline fitting method disclosed in U.S. Pat. No. 5,983,162, and a straight-line fitting method disclosed in U.S. Pat. No. 6,990,436, for instance.
When the data to be analyzed is two-dimensional, the well-known envelope-function construction methods include U.S. Pat. No. 6,311,130 proposed by Huang N. E. in which two-dimensional signal is regarded as a combination of one-dimensional signals, the empirical mode decomposition is used to decompose the one-dimensional signals and the decomposed signals are then combined into two-dimensional intrinsic mode functions. Y. Xu et al. employ triangular grids to construct the required envelope surface in the sifting procedure. Nunes J. C. et al. adopt radial basis function interpolation to accomplish the envelope-plane construction. The method of using grids obtains better results but is not applicable to three-dimensional empirical mode analysis. Theoretically, the interpolation method can be applied to three-dimensional applications, but has a poor continuality on space differential. Besides, a researcher Per Cloersen has proposed US Patent No. 2002/0186895, which analyzes the fluctuation of the empirical mode analysis along with time axis. In spite of the above prior-art disclosures, there is still no method for analyzing a three-dimensional empirical mode at present.
Therefore, owing that the present image-analysis application, such as medical image analysis or other application and scientific research have entered a three-dimensional field, the prior-art technology is insufficient for the requirement in the three-dimensional data analysis. For this reason, it becomes a topic to develop an algorithm of three or more than three dimensional empirical mode decomposition.